1984 Mercury Sable Battery vs Alternator

Hi!

My Mercury Sable is my backup car, and sometimes sits as long as a week without use. I live in CT and the late winters can be cold.

My battery is less than a year old, and recently I tried to start my car and it wouldn't turn over, after almost catching. I jump started it with cables, it started almost right away, I ran it on the highway for 30 minutes, and it was fine. Drove it a couple of times in the next day or so, and it started right away.

It then sat for another 5 days without being started, and wouldn't turn over. Seems like the battery just drained, even though the weather wasn't very cold over those 5 days.

The battery is under warranty, but I believe my alternator is fine, since it jumped ok and ran as I described.

My fear is that the garage that sold me the battery under full warranty will try to tell me I need a new alternator instead of replacing the battery for free.

What are my options, and is there someway from what I describe that you can tell its the battery and not the alternator?

Thanks!
Jerry